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Theorem isusp 22870
Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
isusp.1 𝐵 = (Base‘𝑊)
isusp.2 𝑈 = (UnifSt‘𝑊)
isusp.3 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
isusp (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3498 . 2 (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2 0nep0 5245 . . . . 5 ∅ ≠ {∅}
3 isusp.1 . . . . . . . . . . . 12 𝐵 = (Base‘𝑊)
4 fvprc 6654 . . . . . . . . . . . 12 𝑊 ∈ V → (Base‘𝑊) = ∅)
53, 4syl5eq 2871 . . . . . . . . . . 11 𝑊 ∈ V → 𝐵 = ∅)
65fveq2d 6665 . . . . . . . . . 10 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7 ust0 22828 . . . . . . . . . 10 (UnifOn‘∅) = {{∅}}
86, 7syl6eq 2875 . . . . . . . . 9 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
98eleq2d 2901 . . . . . . . 8 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10 isusp.2 . . . . . . . . . 10 𝑈 = (UnifSt‘𝑊)
1110fvexi 6675 . . . . . . . . 9 𝑈 ∈ V
1211elsn 4565 . . . . . . . 8 (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
139, 12syl6bb 290 . . . . . . 7 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14 fvprc 6654 . . . . . . . . 9 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1510, 14syl5eq 2871 . . . . . . . 8 𝑊 ∈ V → 𝑈 = ∅)
1615eqeq1d 2826 . . . . . . 7 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
1713, 16bitrd 282 . . . . . 6 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
1817necon3bbid 3051 . . . . 5 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
192, 18mpbiri 261 . . . 4 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
2019con4i 114 . . 3 (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
2120adantr 484 . 2 ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22 fveq2 6661 . . . . . 6 (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
2322, 10syl6eqr 2877 . . . . 5 (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24 fveq2 6661 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
2524, 3syl6eqr 2877 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2625fveq2d 6665 . . . . 5 (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
2723, 26eleq12d 2910 . . . 4 (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28 fveq2 6661 . . . . . 6 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29 isusp.3 . . . . . 6 𝐽 = (TopOpen‘𝑊)
3028, 29syl6eqr 2877 . . . . 5 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
3123fveq2d 6665 . . . . 5 (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
3230, 31eqeq12d 2840 . . . 4 (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
3327, 32anbi12d 633 . . 3 (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34 df-usp 22866 . . 3 UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
3533, 34elab2g 3654 . 2 (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
361, 21, 35pm5.21nii 383 1 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  Vcvv 3480  c0 4276  {csn 4550  cfv 6343  Basecbs 16483  TopOpenctopn 16695  UnifOncust 22808  unifTopcutop 22839  UnifStcuss 22862  UnifSpcusp 22863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-ust 22809  df-usp 22866
This theorem is referenced by:  ressust  22873  ressusp  22874  tususp  22881  uspreg  22883  ucncn  22894  neipcfilu  22905  ucnextcn  22913  xmsusp  23179
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